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Chapter 8: Problem 7

Suppose that you decide to buy a car for \(\$ 29,635\), including taxes and license fees. You saved \(\$ 9000\) for a down payment and can get a five-year car loan at \(6.62 \%\). Find the monthly payment and the total interest for the loan.

Short Answer

Expert verified
The monthly payment for the car loan is approximately \$401.38 and the total interest paid over the course of the loan is approximately \$4,083.

Step by step solution

01

Calculate the loan amount

Subtract the down payment from the total cost of the car. The equation will be: Loan Amount = Total Cost - Down Payment. Therefore, Loan Amount = \$29,635 - \$9000 = \$20,635.
02

Convert Annual Interest Rate to Monthly Interest Rate

Convert the annual interest rate of 6.62% into a monthly interest rate. Divide the annual rate by 12 (the number of months in a year). The monthly interest rate is then \(6.62\% / 12 = 0.5517\% \) per month or \(0.005517\) in decimal form.
03

Determine the Loan Term in Months

The loan term is given as 5 years, but we are trying to find the monthly payments, so the term must be in months. Multiply the number of years by 12 (the number of months in a year). Therefore, the loan term is \(5 * 12 = 60\) months.
04

Calculate the Monthly Payment

Use the formula for calculating the monthly payment on a loan, which is \(P = \[PV * r * (1 + r)^n\] / \[(1 + r)^n - 1]\), where - P is the monthly payment - PV is the loan amount - r is the monthly interest rate - n is the number of payments (or the loan term in months). Substituting the values, Monthly Payment = \[\$20,635 * 0.005517 * (1 + 0.005517)^{60}\] / \[(1 + 0.005517)^{60} - 1]\]
05

Calculate the Total Interest

To calculate the total interest paid over the life of the loan, multiply the monthly payment by the number of payments and then subtract the original loan amount from this result. Total Interest = (Monthly Payment * Term) - Loan Amount.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Monthly Payment Calculation
When buying a big-ticket item like a car, understanding how your monthly payments are calculated can make a huge difference in managing your finances effectively.
To determine these payments, we rely on a specific loan payment formula. This formula takes into account factors such as the initial loan amount, the monthly interest rate, and the total number of payments over the loan's term.

The formula used is:
- \[ P = \frac{PV \times r \times (1 + r)^n}{(1 + r)^n - 1} \] where:
  • \(P\) is the generated monthly payment amount.
  • \(PV\) stands for the present value, or the loan amount.
  • \(r\) represents the monthly interest rate (expressed as a decimal).
  • \(n\) is the total number of payments, corresponding to the loan term in months.
By inserting these relevant values into the formula, we generate a monthly payment that will remain consistent over the duration of the loan.
Interest Rate Conversion
Interest rates can be quoted in various forms, but the most common ones revolve around annual percentages. However, when calculating monthly payments for loans, it's crucial to express these interest rates on a monthly basis.
Converting an annual interest rate into a monthly rate requires dividing by 12 鈥� the number of months in a year.

For聽instance, if you have an annual rate of 6.62%, it needs to be converted into this equation:
  • The monthly interest rate = \( \frac{6.62\%}{12} = 0.5517\% \)
  • In decimal form, this becomes \(0.005517\) per month.
This adjustment ensures that the interest cost is accurately distributed each month.
Loan Term Calculation
The term of a loan refers to the length of time over which a borrower is expected to pay back the borrowed amount. For most loans, this period is expressed in years. However, for monthly payment calculations, it's more practical to convert this term into months.
Converting the term is straightforward: multiply the number of years by 12.

For example, a 5-year loan term corresponds to:
  • The monthly equivalent = \(5 \times 12 = 60\) months.
This conversion allows us to input the correct period (in terms of payments) into our monthly payment calculation formula.
Total Interest Calculation
Finally, understanding the total interest you will pay over the life of your loan is crucial for financial planning. It reveals the true cost of borrowing beyond just the principal amount.
To compute the total interest, follow these steps:
  • First, determine the total amount paid over the loan by multiplying the monthly payment by the number of payments.
  • Subtract the original loan amount from this product.
The formula looks like this:\[ \text{Total Interest} = (\text{Monthly Payment} \times \text{Loan Term}) - \text{Loan Amount} \]
By knowing the total interest, you can evaluate whether the loan's terms are favorable compared to other potential options.

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Most popular questions from this chapter

The price of a condominium is \(\$ 180,000\). The bank requires a \(5 \%\) down payment and one point at the time of closing. The cost of the condominium is financed with a 30 -year fixed-rate mortgage at \(8 \%\). a. Find the required down payment. b. Find the amount of the mortgage. c. How much must be paid for the one point at closing? d. Find the monthly payment (excluding escrowed taxes and insurance). e. Find the total cost of interest over 30 years.

To borrow money, you pawn your mountain bike. Based on the value of the bike, the pawnbroker loans you \(\$ 552\). One month later, you get the bike back by paying the pawnbroker \(\$ 851\). What annual interest rate did you pay?

In Exercises 3-4, find the gross income, the adjusted gross income, and the taxable income. Base the taxable income on the greater of a standard deduction or an itemized deduction. Suppose your neighbor earned wages of \(\$ 86,250\), received \(\$ 1240\) in interest from a savings account, and contributed \(\$ 2200\) to a tax-deferred retirement plan. She is entitled to a personal exemption of \(\$ 3800\) and a standard deduction of \(\$ 5950\). The interest on her home mortgage was \(\$ 8900\), she contributed \(\$ 2400\) to charity, and she paid \(\$ 1725\) in state taxes.

How much money should be deposited today in an account that earns \(6 \%\) compounded semiannually so that it will accumulate to \(\$ 10,000\) in three years?

Suppose that you are buying a car for \(\$ 56,000\), including taxes and license fees. You saved \(\$ 8000\) for a down payment. The dealer is offering you two incentives: Incentive \(\mathrm{A}\) is \(\$ 10,000\) off the price of the car, followed by a four-year loan at \(12.5 \%\). Incentive \(\mathrm{B}\) does not have a cash rebate, but provides free financing (no interest) over four years. What is the difference in monthly payments between the two offers? Which incentive is the better deal?
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